Question

Solve the following equations by the substitution method :

$$\begin{gathered} \left(x\right)\frac{(3x-y)}{5}=2y-1 \\ \frac{3x}{8}-\frac{y}{4}=\frac{1}{2} \end{gathered}$$

Answer 1

Step 1: Express one variable (say x) in terms of the other variable (say y) from one of the given equations:

$\frac{3x-y}{5}=2y-1$

Multiplying the equation throughout by 5

$$\begin{gathered} 3x-y=5(2y-1) \\ 3x-y=10y-5 \\ 3x=10y+y-5 \\ 3x=11y-5 \\ x=\frac{11y-5}{3} \end{gathered}$$

Step 2: Substitute this value of x in the other equations to get a linear equation in y which can be solved:

$\frac{3x}{8}-\frac{y}{4}=\frac{1}{2}$

Multiplying the equation throughout by 8:

$$\begin{gathered} 3x-2y=4 \\ 3\left(\frac{11y-5}{3}\right)-2y=4 \\ 11y-5-2y=4 \\ 9y-5=4 \\ 9y=4+5 \\ 9y=9 \\ y=\frac{9}{9} \\ y=1 \end{gathered}$$

Step 3: Substitute the values of y obtained in $x=\frac{11y-5}{3}$, to get the values of x:

$$\begin{gathered} x=\frac{11y-5}{3} \\ x=\frac{(11\times 1)-5}{3} \\ x=\frac{11-5}{3} \\ x=\frac{6}{3} \\ x=2 \end{gathered}$$

Hence, the value of $x=2andy=1$.